Let's go through the problem step-by-step.

Problem Recap:

We're given that the noise level $S(d)$ in decibels, as a function of distance $d$ (in meters) from a wind turbine, can be modeled by the function: $S(d) = ab^d$ We know two specific points:

1. When $d = 0$, $S(0) = 105$ decibels.
2. When $d = 100$, $S(100) = 49$ decibels.

We need to:

1. Use this information to set up two equations for $a$ and $b$.
2. Solve for $a$ and $b$.

Solution:

Step 1: Set Up Equations

1. For $d = 0$: $S(0) = ab^0 = a \cdot 1 = a$ Since $S(0) = 105$, we get: $a = 105$

2. For $d = 100$: $S(100) = ab^{100} = 105 \cdot b^{100}$ Since $S(100) = 49$, we can write: $105 \cdot b^{100} = 49$

Step 2: Solve for $b$

From the equation $105 \cdot b^{100} = 49$, isolate $b$ by dividing both sides by 105: $b^{100} = \frac{49}{105}$ Simplify the fraction: $b^{100} = \frac{7}{15}$ Now, take the 100th root of both sides to solve for $b$: $b = \left(\frac{7}{15}\right)^{\frac{1}{100}}$

This gives us the values of $a$ and $b$:

• $a = 105$
• $b = \left(\frac{7}{15}\right)^{\frac{1}{100}}$

Would you like a further breakdown of these steps or have additional questions on this solution?

Follow-up Questions:

1. How can the value of $b$ be approximated numerically for practical purposes?
2. What happens to the noise level $S(d)$ as $d$ increases indefinitely?
3. How would the model change if we wanted a different initial decibel level at $d = 0$?
4. Can we determine a maximum distance where the noise level falls below a certain threshold using this model?
5. How does the choice of an exponential model $S(d) = ab^d$ reflect the real-world behavior of sound attenuation?

Tip:

In exponential models like this, the base $b$ controls the rate of decay (or growth). A smaller base leads to a faster decrease in the function's values as $d$ increases.